Parameter selection rules for path-following with singular embedding (Q1805868)

Embedding methods form one of the basic tools for the numerical treatment of operator equations and inequalities. While regular embeddings have dominated the literature in the past, the research in embedding techniques has shifted to singular ones where bifurcations occur or to ill-posed limit problems. In the present paper firstly quite general parameter selection rules are proposed which guarantee, under weak conditions, the overall convergence of an implementable iteration process arising from a locally quadratically convergent method truncated at each level of the embedding process combined with a trivial predictor. Secondly, some modifications in the case of polynomial error and contraction bounds are studied. Finally, these results are applied to the embedding of an elliptic boundary value problem with discontinuous nonlinearities into a family of smooth problems. Here the regularization is done in such a way that the solutions of the resulting auxiliary problems require only one step of Newton's method.